New York Journal of Mathematics
New York J. Math. 14 (2008) 459–494.
Small time heat kernel behavior on
Riemannian complexes
Melanie Pivarski and Laurent Saloff-Coste
Abstract. We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These
inequalities have a variety of applications, including bounds on the heat
kernel and a uniform local Harnack inequality. We additionally consider
the example of a complex, X, which has a finitely generated group of
isomorphisms, G, such that X/G = Y is a complex consisting of a finite
number of polytopes. We show that when this group, G, has polynomial volume growth, there is a uniform global Poincaré inequality on
the complex, X.
Contents
1.
Introduction
1.1. Definitions
1.2. The Dirichlet form
2. Poincaré inequalities
3. Applications
3.1. Heat kernel bounds
3.2. Groups with polynomial growth
3.3. Further remarks
References
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Received March 28, 2008.
Mathematics Subject Classification. 26D10, 35B40, 43A85, 57S, 58J35.
Key words and phrases. Poincaré inequality, heat kernel, heat equation, polynomial
growth group, Euclidean complex, Riemannian complex, polytopal complex, polyhedral
complex.
Pivarski’s research was partially supported by NSF Grant DMS 0306194; Saloff-Coste’s
research was partially supported by NSF Grant DMS 0603886.
ISSN 1076-9803/08
459
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Melanie Pivarski and Laurent Saloff-Coste
1. Introduction
It has been observed by several authors, see, e.g., [20, 33], that a theory
called first-order calculus can be developed under some assumptions on metric measure spaces. The heat equation and its associated Markov process,
Brownian motion, require some additional structure. For instance, it is well
understood that the structure of complete Riemannian manifolds induces a
well-defined heat equation and Brownian motion. Similarly, it is natural to
ask if the structure of Riemannian polyhedral complexes yields a well-defined
notion of heat equation and Brownian motion. For one-dimensional complexes (i.e., locally finite metric graphs), this has been studied by probabilists
under the name of Walsh Brownian motion. Strictly speaking, Walsh Brownian motion is defined on a (perhaps finite) collection of semiaxes with the
same origin. See, e.g., [5, 4]. A construction in this spirit on 2-dimensional
Euclidean simplicial complexes is given in [8, 14]. For more general complexes, a completely different approach is considered in [7]. Except in dimension 1, the question of the unicity of the constructed objects has not
been thoroughly studied and presents some difficulties. In this paper, we
define the heat equation (and, implicitly, the associated process) using the
Dirichlet form approach as in [13]. Indeed, just as a Riemannian manifold
carries a natural Dirichlet form, so does a Riemannian polyhedral complex.
Under certain assumptions, we prove basic estimates for the associated heat
kernel.
Riemannian polyhedral complexes are formed by taking a collection of
n-dimensional convex polyhedra and joining them along n − 1 dimensional
faces. Within each polyhedron, we will have the same metric structure as
a Riemannian manifold. When we join them, we will glue the faces of two
polyhedra together so that points on one face are identified with points on
the other face, and the metrics on those faces are preserved. We will require
that these structures have an upper bound on the number of n-dimensional
polyhedra that share an n − 1-dimensional face and a lower bound on the
interior angles and distances between nonadjacent faces. Additionally we
assume a bound on the ellipticity of the manifolds. The complex formed
by looking at k-dimensional faces is called the k-skeleton. For instance,
the 0-skeleton is set of vertices. A 1-skeleton is a graph where the space
includes both vertices and points on the edges; sometimes this is called a
metric graph [27]. Note that we can triangulate any convex polyhedron to
obtain a collection of simplices, and so when the metrics are all Euclidean,
this structure is essentially equivalent to looking at a simplicial complex. In
Section 1.1 we define these structures as well as some restrictions on their
geometry. In Section 1.2 we define and describe the Dirichlet form and its
domain.
In Section 2, we show a Poincaré inequality on such a complex, X: For
any fixed R0 , there exists a constant, P0 , so that for any r < R0 , z ∈ X,
Small time heat kernel behavior
and f ∈ W 1,p (B(z, r)) we have
p
p
|f (x) − fB(z,r) | dμ(x) ≤ P0 r
B(z,r)
461
|∇f (x)|p dμ(x).
B(z,r)
This inequality allows us to describe small time heat kernel behavior;
we do so in Section 3.1. Many important properties follow. Theorems
in Sturm [38] can be applied directly to these complexes to show that on
any compact subset of the k-skeleton, the heat kernel is locally like the
one on Rk , with constants that depend on the choice of compact subset.
The essential improvement in our theorem is that the constants are uniform
throughout the entire complex. Results where there are only a finite number
of glued spaces can be found in Paulik [29] who additionally studies sets
whose overlap has positive measure. In Section 3.2 we consider a complex,
X, which has a finitely generated group of isomorphisms, G, such that X/G
is a complex consisting of a finite number of polytopes. We show that for a
group, G, with polynomial volume growth, there is a uniform global Poincaré
inequality on the complex, X. In this case, the heat kernel asymptotics apply
with global constants.
1.1. Definitions. We begin by defining the complex and its geometric
structure, as well as some restrictions on this structure. For a thorough
introduction to analysis on polyhedral complexes, see [13].
Definition 1.1. A polyhedral complex X is the union of a collection of
convex polyhedra which are joined along lower-dimensional faces. By this we
mean that for any two distinct polyhedra P1 , P2 in the collection, P1 ∩P2 is a
polyhedron whose dimension satisfies dim(P1 ∩P2 ) < max(dim(P1 ), dim(P2 ))
and P1 ∩ P2 is a face of both P1 and P2 . We allow this face to be the empty
set.
Note that this definition implies P1 ∩ P2 is a connected set. This rules out
expressing a circle as two edges whose ends are joined, but it allows us to
write it as a triangle of three edges. This definition is not very restrictive, as
we can triangulate the polyhedra in order to form a complex which avoids
the overlap.
Simplicial complexes are an example of a polyhedral complex; the difference here is that we allow greater numbers of sides. Note that we allow
unbounded polyhedra, not just bounded polytopes. We do not define or
require an embedding of the complex into Euclidean space; however, each
individual polytope or polyhedron can be viewed locally as a subset of Euclidean space.
Definition 1.2. Define a k-skeleton, X (k) , for 0 ≤ k ≤ dim(X) to be
the union of all faces of dimension k or smaller. Note that this is also a
polyhedral complex.
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Melanie Pivarski and Laurent Saloff-Coste
Figure 1. Example of a two-dimensional Euclidean complex
(left) and its 1-skeleton (right).
Figure 2. Examples of a complex which is not dimensionally homogeneous (left), one which is not 1-chainable (center), and one which is admissible (right).
Definition 1.3. A maximal polyhedron is a polyhedron that is not a proper
face of any other polyhedron. We say X is dimensionally homogeneous if
all of its maximal polyhedra have dimension n. Note that in combinatorics
literature this is called pure. We denote the set of maximal polyhedra by
M.
Definition 1.4. X is locally (n − 1)-chainable if for every connected open
set U ⊂ X, U − X (n−2) is also connected. For a dimensionally homogeneous
complex X this is equivalent to the property that any two n-dimensional
polyhedra that share a lower-dimensional face can be joined by a chain
of contiguous (n − 1) or n-dimensional polyhedra containing the lowerdimensional face.
Definition 1.5. We call X admissible if it is dimensionally homogeneous
and in some triangulation X is locally (n − 1)-chainable. (See Figure 2.)
We will be working with connected admissible complexes. On each maximal polyhedron, P , we have a covariant bounded measurable Riemannian
metric tensor G which satisfies an ellipticity condition. There exists a constant, ΛP , so that for any ζ ∈ Rn we have
(ζ i )2 ≤ Gij ζ i ζ j ≤ Λ2P
(ζ i )2 .
Λ−2
P
We require the metric to be continuous in the sense that G is continuous
up to the boundary, and the metrics on two neighboring polyhedra induce
the same Riemannian metric on their shared face. We will also require the
ellipticity to be uniform; that is,
Λ = sup ΛP < ∞.
P ∈M
Small time heat kernel behavior
463
Figure 3. Complex with shaded ball B (left); the three
wedges for B (right).
Distance is defined as in [13] as the infimum over a set of lengths of paths.
Note that this is an intrinsic distance, so X is a length space.
We will set the
Let X = ∪i Pi , where the Pi are the maximal polyhedra.
measure of A, a Borel subset of X, to be μ(A) = i μi (A ∩ Pi ) where μi is
the measure on Pi . We will use the notation μe and μg whenever we need
to distinguish between the Euclidean and Riemannian measures.
Definition 1.6. An admissible polyhedral complex, X, equipped with a
uniformly elliptic Riemannian metric tensor G on each polyhedron is called a
Riemannian polyhedral complex. When Λ = 1, it is a Euclidean polyhedral
complex. For brevity, we will often call this a Riemannian (respectively
Euclidean) complex.
The uniform ellipticity condition forces each Riemannian complex to have
a corresponding Euclidean complex with comparable distance:
Λ−1 dg ≤ de ≤ Λdg .
Similarly, for a complex whose maximal polyhedra have dimension n, the
measures are comparable: Λ−n μg ≤ μe ≤ Λn μg .
Definition 1.7. Let X be an admissible polyhedral complex of dimension
n such that for every k ≤ n the distance between any two nonintersecting
k-dimensional faces is bounded below. Let B be a Euclidean ball of radius r
whose center is on a D-dimensional face with the property that B intersects
no other D-dimensional faces. We define wedges Wj of B to be the closures
of each of the connected components of B − X (n−1) .
Note that for any z in such an X, a ball B(z, r) satisfying the above
criteria exists: for each D, we can take any point z ∈ X (D) − X (D−1) and
any r < d(z, X (D−1) ) and create B = B(z, r) ⊂ X. Then B is a ball
of radius r whose center is on a D-dimensional face, and B intersects no
other D-dimensional faces. In essence, the wedges, Wj , are formed when
the (n − 1)- skeleton slices the ball B into pieces. Each Wj has diameter
at most 2r, as each of the points in Wj is within distance r of z, and z is
included in Wj .
Example 1.8. In Figure 3 we have an example of a 2-dimensional complex
with a shaded ball centered at a vertex. This ball has three wedges; one for
each of the two-dimensional faces that share the vertex. Each wedge is a
fraction of a sphere.
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Melanie Pivarski and Laurent Saloff-Coste
Definition 1.9. We say X has solid angle bound α if, with respect to
Euclidean distance and volume, for z ∈ X (D) −X (D−1) and r < d(z, X (D−1) )
the wedges of the ball B(z, r) satisfy
α≤
μ(Wj )
≤ 1.
μ(r n S (n−1) )
Note that the right-hand side of the inequality reflects the fact that each
of the Wj is a subset of a Euclidean ball.
Assumptions 1.10. We require X to satisfy the following geometric assumptions:
(1) X is a connected admissible complex with n-dimensional maximal
polyhedra.
(2) X is uniformly elliptic with constant Λ.
(3) For every k, the maximal number of faces in X (k) that can share a
lower-dimensional face is bounded above by M .
(4) For every k, the distance between any two nonintersecting k-dimensional faces is bounded below by .
(5) X has solid angle bound α.
Note that assumption (3) implies every vertex has degree at most M .
Similarly, assumption (4) implies edge lengths are bounded below by as
vertices are 0-dimensional faces.
These assumptions imply each closed ball of finite radius will intersect
only finitely many polyhedra. As each of these intersections forms a closed
bounded subset of a polyhedron, and each of these is complete, X is complete.
Under Assumptions 1.10, volume doubling occurs locally with a uniform
constant. For any R, there exists a constant c so that for any x ∈ X and
r < R,
μ(B(x, 2r)) ≤ cμ(B(x, r)).
Note that volume doubling will not necessarily hold globally.
Notation1.11. Lp norms
1/p restricted to a subset A ⊂ X are written as
.
||f ||p,A = A |f (x)|p dμ
1.2. The Dirichlet form. Now that we have defined the space geometrically, we will define a Dirichlet form whose core consists of compactly
supported Lipschitz functions. We denote the space of Lipschitz functions by Lip(X) and the space of compactly supported Lipschitz functions
by C0Lip (X). Note that Lipschitz functions are continuous and differentiable almost everywhere. By Theorem 4 in Section 5.8 of [15], for each
B(x, ) ⊂ X − X (n−1) and f ∈ C0Lip (X), f restricted to B(x, ) is in the
Sobolev space W 1,∞ (B(x, )). This tells us that f has a gradient almost
everywhere in X − X (n−1) . Since μ(X (n−1) ) = 0, f has a gradient for almost
every x in X.
Small time heat kernel behavior
465
We would like an energy form that acts like E(u, v) = X ∇u, ∇v
dμ for
u, v in its domain, Dom(E), to define the operator Δ with domain Dom(Δ).
We can define E in a very general manner which does not depend on the
local structure by following a paper of Sturm [39]. We can also define it in
a more straightforward manner which uses the geometry of X. We do both,
and then show that they coincide.
Sturm assumes that the space (X, d) is a locally compact separable metric
space, μ is a Radon measure on X, and that μ(U ) > 0 for every nonempty
open set U ⊂ X. By 1.10, these assumptions hold both in X and on the
skeleta, X (k) .
Definition 1.12. We define E r as
E r (u, v)
=
X
B(x,r)−{x}
2ndμ(y)dμ(x)
(u(x) − u(y))(v(x) − v(y))
2
d (x, y)
μ(B(x, r)) + μ(B(y, r))
for u, v ∈ Lip(X) where n is the local dimension.
Each E r with domain C0Lip (X) is closable and symmetric on L2 (X), and
its closure has core C0Lip (X). See Lemma 3.1 in [39]. One can take limits of
these Dirichlet forms in the following way. The Γ-limit of the E rn is defined
to be the limit that occurs when the following lim sup and lim inf are equal
for all u ∈ L2 (X). See Dal Maso [11] for a thorough treatment.
Γ − lim sup E rn (u, u) := lim lim sup
n→∞
α→0 n→∞
Γ − lim inf E rn (u, u) := lim lim inf
n→∞
inf
E rn (v, v)
inf
E rn (v, v).
v∈L2 (X)
||u−v||≤α
α→0 n→∞ v∈L2 (X)
||u−v||≤α
For any sequence {E rn } with rn → 0 , there is a subsequence {rn } so that
the Γ-limit of E rn exists by Lemma 4.4 in [39]. These lemmas are put together in Theorem 5.5 in [39] which tells us that this limit, E 0 , with domain
C0Lip (X) is a closable and symmetric form, and its closure, (E, C0Lip (X)), is
a strongly local regular Dirichlet form on L2 (X) with core C0Lip (X).
Alternatively, we can define the energy form using the structure of the
space.
Definition 1.13. For f ∈ C0Lip (X) we set E0 (·, ·) to the following:
|∇f |2 dμ.
E0 (f, f ) =
P ∈M P
Lemma 1.14. (E0 (·, ·), C0Lip (X)) is a closable form.
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Melanie Pivarski and Laurent Saloff-Coste
Lip
Proof. To show this, we must prove that any sequence {fn }∞
n=1 ⊂ C0 (X)
that converges to 0 in L2 (X) and is Cauchy in || · ||2 + E(·, ·) satisfies
limn→∞ E(fn , fn ) = 0. We will first look at what happens on one fixed
polyhedron and then look at what happens on the complex. Let P be a
maximal polyhedron. Since {fn }∞
n=1 is Cauchy in the norm, we have
1 1
2
2
2
2
(fn − fm ) dμ
+
(∇fn − ∇fm ) dμ
= 0.
lim
m,n→∞
P
P
This gives us two functions, f and F which are the limits of fn and ∇fn
respectively. We have f = 0 by assumption. On the interior of P , we
have theusual Dirichlet form; this implies F = 0 on the interior of P and
limn→∞ P ∇fn dμ = 0. Since μ(X − X (n−1) ) = 0, F = 0 a.e. on X.
To show L2 convergence, we need to interchange the limit with the sum
over the maximal polyhedra. We can do this for |∇fn − ∇fm | by Fatou’s
Lemma.
|∇fn |2 dμ = lim
|∇fn − lim ∇fm |2 dμ
lim
n→∞
P ∈M P
n→∞
= lim
n→∞
m→∞
P ∈M P
lim |∇fn − ∇fm |2 dμ
m→∞
P ∈M P
≤ lim lim
n→∞ m→∞
|∇fn − ∇fm |2 dμ
P ∈M P
= 0.
This tells us that the form is closable.
We will show that the two energy forms, E and the closure of E0 , are the
same. To do this, we show that they are the same on the core C0Lip (X).
Lemma 1.15. Let (E, Dom(E)) be the closure of (E0 , C0Lip (X)). Then
Dom(E) = Dom(E),
and for any f ∈ Dom(E) we have E(f, f ) = E(f, f ).
Proof. We can write X as (X − X (n−1) ) ∪ X (n−1) ; this is a collection of
maximal polyhedra and a set of measure 0. The interior of each maximal
polyhedron is a Riemannian manifold without boundary. X is also a locally
compact length space, and so it satisfies the conditions of Example 4G in
[38]. This implies it has the strong measure contraction property with an
exceptional set. Corollary 5.7 in [38] tells us E(f, f ) = E(f, f ) for each
f ∈ C0Lip (X). The equality is shown by approximating the forms using an
increasing sequence of open subsets which limit to X − X (n−1) . As C0Lip (X)
is a core for both E and E, the Dirichlet forms are the same.
Note that this equality tells us that the forms E rn have a unique Γ−limit.
Our next result describes what the domain of this form is in more concrete
terms.
Small time heat kernel behavior
467
Definition 1.16. We define
W 1,2 (X) = {f ∈ L2 (X) | for any maximal polyhedron, P, f |P ∈ W 1,2 (P ),
E(f, f ) < ∞, and Tri (f ) = Trj (f ) on Pi ∩ Pj }
where Tri : W 1,2 (Pi ) → L2 (∂Pi ) is a trace function on the maximal polyhedron Pi . We write W01,2 (X) for the set of compactly supported functions in
W 1,2 (X). Similarly, for any domain Ω ⊂ X and 1 ≤ p < ∞ we set
W 1,p (Ω)
=
f ∈ Lp (Ω) | for any maximal polyhedron, P, f |P ∩Ω ∈ W 1,p (P ∩ Ω),
|∇f | dμ < ∞, and Tri (f ) = Trj (f ) on Pi ∩ Pj ∩ Ω
p
P ∈M P ∩Ω
where Tri : W 1,p (Pi ) → Lp (∂Pi ) is a trace function on the maximal polyhedron Pi .
Theorem 1.17. Dom(E) = W 1,2 (X).
Proof. We know that Dom(E) = C0Lip (X), where the closure is taken with
respect to the W 1,2 (X) norm, ||·||W 1,2 = ||·||2 +E(·, ·). For any f ∈ C0Lip (X),
the support of f is a set with finite measure. For any maximal polyhedron,
P , f restricted to P will be in W 1,2 (P ), since
||∇f ||2,P ≤ ||∇f ||∞,P μ(P ∩ supp(f )).
Similarly, since f has compact support, E(f, f ) < ∞.
We can view each maximal polyhedron as a subset of an n dimensional
manifold. Then each f ∈ Dom(E) has the property that f |P is in W 1,2 (P ).
In particular, polyhedra are Lipschitz domains, and so we can apply Theorem 1.12.2 from Chapter 14 of [13] to these maximal polyhedra. This theorem tells us that for each maximal polyhedron, Pi , a well-defined trace function, Tri : W 1,2 (Pi ) → L2 (∂Pi ) exists. In particular, the trace is a bounded
linear operator, and so for continuous functions, Tri (f |Pi ) = f |Pi . This gives
us C0Lip (X) ⊂ W 1,2 (X). When f is the limit of functions fm ∈ C0Lip (X),
Tri (f |Pi ) is the limit of Tri (fm |Pi ). For every pair of maximal polyhedra, Pi
and Pj , we have:
Tri f |Pi ∩Pj = lim Tri fm |Pi ∩Pj
m→∞
= lim fm |Pi ∩Pj
m→∞
= lim Trj fm |Pi ∩Pj
m→∞
= Trj f |Pi ∩Pj .
Thus for every f ∈ C0Lip (X) we have Tri = Trj on Pi ∩ Pj . This shows
that C0Lip (X) ⊂ W 1,2 (X). We will now show the reverse containment.
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Melanie Pivarski and Laurent Saloff-Coste
We choose an arbitrary point in X and let Bn represent the ball of radius n
centered at that point. Due to the geometric assumptions on X, Bn will have
finite volume. Suppose that f is in the set W 1,2 (X). We can approximate
f in the W 1,2 (X) norm with a sequence of compactly supported functions,
fn , in W01,2 (X) which have the property that the support of fn is contained
in B2n and fn = f on Bn .
Theorem 4.1 in Shanmugalingam [34], says that under certain conditions
on the space, fn can be approximated in the W 1,2 (B2n+1 ) norm by a sequence of locally Lipschitz functions, hn,k with compact support in B2n+1 .
Assume for the moment that these conditions hold. By a diagonal argument
hn,n tends to f in W 1,2 (X). This shows that f ∈ C0Lip (X).
To complete the proof, we need only show that B2n+1 satisfies the conditions of Theorem 4.1 in Shanmugalingam [34]. To do so, we need the
following concept.
Definition 1.18. Let u be a real valued function and ρ be a nonnegative
Borel measurable function which satisfies the following inequality for all
compact rectifiable paths γ with endpoints x and y:
|u(x) − u(y)| ≤ ρds.
γ
The function ρ is called an upper gradient (or very weak gradient) of u. See,
1,1
(X), |∇u|
e.g., [21] for a discussion of such functions. Note that if u ∈ Wloc
is an upper gradient for u.
The conditions of Theorem 4.1 in Shanmugalingam [34] are as follows.
The first is that volume doubling holds in the ball B2n+1 ; as noted earlier,
this holds. The second condition is that all pairs of measurable functions and
their upper gradients (u, ρ) satisfy the following Poincaré style inequality for
λ = 1 and p = 2.
1/p
p
.
− |u − uB |dμ ≤ C diam(B) − ρ dμ
B
λB
Here B is any ball contained in B2n+1 . C does not depend on B, though it
does depend on B2n+1 .
By Theorem 6.11 in [21], when we consider an individual polyhedral subset of B2n+1 , the Poincaré style inequality will hold for all Lipschitz functions, u, and their upper gradients for some λ ≥ 1 and p = 1. Since X
is chainable, we can form the entire set by gluing together finitely many
pieces with sufficient overlap. Theorem 6.15 in [21] says that the glued set
satisfies the Poincaré style inequality for all Lipschitz functions as well. In
[22], Heinonen and Koskela show that we can replace the condition that u
is Lipschitz with the condition that u is measurable. We switch from λ ≥ 1
to λ = 1 by using Whitney covers; the argument in Section 5.3 of [31] holds
in this case. We switch from p = 1 to p = 2 by Hölder’s inequality.
Small time heat kernel behavior
469
Note that Theorem 4.1 in Shanmugalingam [34] states that Lipschitz
functions are dense in N 1,2 (B2n+1 ), the space of functions and upper gradients, (u, ρ), which have finite norm: ||u||2 + ||ρ||2 < ∞. All gradients are
also upper gradients; in particular, W 1,2 (B2n+1 ) ⊂ N 1,2 (B2n+1 ), and for
u ∈ W 1,2 (B2n+1 ), the norms coincide. So since Lip(B2n+1 ) ⊂ W 1,2 (B2n+1 ),
we also have density in the W 1,2 (B2n+1 ) norm.
Remark 1.19. In the next section, we prove a Poincaré inequality for functions in W 1,2 (B). This inequality can be used in the proof above to show
that C0Lip (X) is dense in W 1,2 (X). This important and nontrivial density
result can be obtained in two rather different ways. One is outlined above
and requires a local Poincaré inequality to be valid for functions in W 1,2 (X).
See also [18, 19]. The idea that the (local) volume doubling and Poincaré
inequality properties imply the density of Lipschitz functions with compact
support in the W 1,2 -norm is useful and important, for instance, in works
concerning analysis on domains in Rn with rough boundary.
Another more specific approach is to show that:
(a) Small neighborhoods of faces of dimension at most n − 2 can be disregarded because they have small capacity.
(b) Any function f ∈ W 1,2 (X) that vanishes in a neighborhood of the
faces of dimension at most n − 2 can be approximated in W 1,2 (X) by
continuous functions that are smooth with bounded derivatives of all
order in each open n-face.
The second part of this line of reasoning requires a specific construction (see,
e.g.,[6]).
The Dirichlet form (E, Dom(E)) on L2 (X) uniquely determines a positive
self-adjoint operator (Δ, Dom(Δ)) on L2 (X). Namely, Dom(Δ) is defined
as the subspace of Dom(E) of those functions v with the property that there
This implies
is a constant C such that E(u, v) ≤ Cu2 for all u ∈ Dom(E).
that there is a function w ∈ L2 (X) such that E(u, v) = X u w dμ and, by
definition, Δv = w. See, e.g., Fukushima, Ōshima, and Takeda [16]. This
sign convention means that when X is the real line, Δf = −f .
It is perhaps useful to emphasize that the Laplacian defined above is an
operator that is rather mysterious.
Definition 1.20. Let D0∞ (X) be the set of all continuous functions with
compact support on X such that the restriction to any open n-face is smooth
with bounded derivatives of all order. Let D = D0∞ (X) ∩ Dom(Δ).
Note that the space D0∞ (X) itself is not contained in Dom(Δ). On
D = D0∞ (X) ∩ Dom(Δ),
Δf is given on each open face by the usual formula in local Euclidean coordinates. However, whether or not the symmetric operator (Δ|D , D) is
essentially self-adjoint on L2 (X) is not known. Nor is it known that the
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Melanie Pivarski and Laurent Saloff-Coste
closure of (Δ|D , D) is (Δ, Dom(Δ)). These real difficulties are easily overlooked. The set D = D0∞ ∩ Dom(Δ) is easy to describe. For any maximal
polyhedron P , let nP be the outward pointing normal unit vector along the
boundary of P .
Proposition 1.21. A function f ∈ D0∞ (X) belongs to Dom(Δ) if and only
if it satisfies
∂f |P
i
= 0 along F
∂
nPi
Pi :F ⊂Pi
for any n − 1-dimensional face F of any maximal polyhedron in X. The set
D is dense in C0 (X) for the uniform norm || · ||∞ and dense in L2 (X).
In this formula, the n − 1-dimensional face F is fixed, and the sum is over
all maximal polyhedra Pi that contain that face (by our assumption, this is
a finite sum). The condition is that, along any fixed n − 1-dimensional face
F , the sum of the outward normal derivatives of the restrictions of f to the
maximal polyhedra meeting along F is zero.
Proof. Because of Theorem 1.17, this easily follows from using the definition
of the Laplacian and Green’s formula on each maximal polyhedron. The fact
that D is dense in C0 (X) easily follows from the fact that, for any compact
set K and for any fixed
small scale, one can construct partitions of unity
covering K, (ωn ), ( n ωn )|K ≡ 1, whose elementary blocks ωn are in D
with each ωn supported in a ball of radius . See [6] for details that easily
generalize to the present situation. Density in L2 (X) follows.
Remark 1.22. Note that this set-up will define a different Laplacian on
each of the k-skeleta. To define E r on a k-skeleton, X (k) , set N = k,
integrate over X (k) , and let μ be a k-dimensional measure. This technique
will define Δk on a dense subset of L2 (X (k) ).
2. Poincaré inequalities
Definition 2.1. We say that f satisfies a weak local p-Poincaré inequality
if there exist constants R0 , κ, and P0 such that
f − fB(x,r) p,B(x,r) ≤ P0 r∇f p,B(x,κr)
holds for all r ≤ R0 , where fB(x,r) is the average of f over B(x, r). If κ = 1,
we say that it is strong. If additionally it holds for all x ∈ X we say that it
is uniform. If R0 = ∞, we say that the inequality is global.
We will show that a uniform local Poincaré inequality holds for complexes
satisfying the geometric assumptions 1.10. Local Poincaré inequalities have
appeared in [21], [42] and [13] for finite complexes or for compact subsets of
complexes. In White’s article [42], a global Poincaré inequality was shown
for Lipschitz functions on an admissible complex made up of a finite number
of polyhedra. The constant in White’s proof is linear in the number of
Small time heat kernel behavior
471
polyhedra involved, and so it does not extend to an infinite complex. A
uniform weak local inequality for Lipschitz functions was also shown on this
finite complex. This too differs from our inequality in its dependence on a
finite complex.
In Eells and Fuglede’s book [13], they show that for any relatively compact
subset of an admissible complex, a local Poincaré inequality will hold for
locally Lipschitz functions with a constant that depends on the particular
choice of compact subset. The larger complex itself can be infinite, but
the constant in the inequality depends on the particular choice of compact
subset.
In Heinonen and Koskela’s article [21], Poincaré inequalities with respect
to the Lq norm are shown for Lipschitz functions with upper gradients on
finite simplicial complexes of pure dimension q with the property that the
link of each vertex is connected. Their approach uses Loewner spaces.
Under the assumptions 1.10 on the geometry of X, we will show there are
constants R0 , P0 ∈ (0, ∞) such that, for any ball B = B(z, r), r < R0 , and
any f ∈ W 1,p (B),
f − fB p,B ≤ pP0 r∇f p,B .
The constants R0 and P0 are constants depending on the space X. Ultimately, for any fixed R0 < ∞, there exists a P0 such that the Poincaré
inequality above holds.
We begin by proving a local Poincaré inequality for an admissible Euclidean polyhedral complex. The Poincaré inequality on a convex subset of
Euclidean space is a well-known statement. We will show it first in a convex
space, and then we will generalize it to our locally nonconvex space.
Notation 2.2. We write the average integral of f over a set A by
fA = − f dx.
A
Lemma 2.3. Let Ω be a connected convex set with Euclidean distance and
structure and Ω1 , Ω2 be n-dimensional convex subsets of Ω, n = dim(Ω).
For f ∈ W 1,1 (Ω), the following holds:
n−1 diam(Ω)
(μ(Ω1 ) + μ(Ω2 )) |∇f (y)|dy.
|f (z) − f (y)|dzdy ≤ 2
n
Ω2 Ω1
Ω
Proof. The type of argument used here is classical and can be found in
many books including Aspects of Sobolev-type inequalities [31]. Details are
included for completeness.
Let γ be a path from z to y. The definition of a gradient gives us:
|f (z) − f (y)| ≤ |∇f (s)|ds.
γ
Note that if we are in a one-dimensional space, a convex subset is a line.
The desired inequality follows from expanding γ to Ω, and then noting that
472
Melanie Pivarski and Laurent Saloff-Coste
integrating over x and y has the effect of multiplying the right-hand side by
μ(Ω1 )μ(Ω2 ) ≤ diam(Ω)(μ(Ω1 ) + μ(Ω2 )).
Because z and y are in the same convex region Ω with Euclidean distance,
we can let the path γ be a straight line:
|y−z| y − z |f (z) − f (y)| ≤
∇f z + ρ |y − z| dρ.
0
We integrate this over z ∈ Ω1 , y ∈ Ω2 . To get a nice bound, we will use a
trick from Korevaar and Schoen [25]. We split the path into two halves. For
each half, we switch into and out of polar coordinates in a way that avoids
integrating 1s near s = 0. This allows us to have a bound which depends on
the volumes of Ω1 and Ω2 rather than Ω.
First, we consider the half of the path which is closer to y ∈ Ω2 . IΩ (·) is
the indicator function for Ω.
|y−z| ∇f z + ρ y − z IΩ z + ρ y − z dρdydz.
|y−z| |y − z| |y − z|
Ω1
Ω2
2
We use a change of variable so that y − z = sθ. That is, |y − z| = s and
y−z
|y−z| = θ. Note that diam(Ω) is an upper bound on the distance between y
and z.
diam(Ω) s
|∇f (z + ρθ)|IΩ (z + ρθ)sn−1 dρdsdθdz.
··· =
S n−1
Ω1
0
s/2
We switch the order of integration. Now, ρ will be between 0 and diam(Ω)
and s will be between ρ and min(2ρ, diam(Ω)). This allows us to integrate
with respect to s.
...
diam(Ω)
min(2ρ,diam(Ω))
=
Ω1
S n−1
0
=
Ω1
S n−1
|∇f (z + ρθ)|IΩ (z + ρθ)sn−1 dsdρdθdz
ρ
diam(Ω)
|∇f (z + ρθ)|IΩ (z + ρθ)
0
(min(2ρ, diam(Ω)))n − ρn
dρdθdz.
n
Now we reverse the change of variables to set y = z + ρθ. Since our integral
includes an indicator function at z + ρθ, we have y ∈ Ω.
(min(2|y − z|, diam(Ω)))n − |y − z|n
|∇f (y)|
dydz.
··· =
n|y − z|n−1
Ω1 Ω
·
n −|y−z|n
One can show (min(2|y−z|,diam(Ω)))
n|y−z|n−1
the z dependence in the integral.
· · · ≤ 2n−1
≤ 2n−1 diam(Ω)
. This will remove
n
diam(Ω)
μ(Ω1 )
n
|∇f (y)|dy.
Ω
Small time heat kernel behavior
473
Figure 4. Complex with shaded ball B (left); the two
wedges for B and a region which overlaps both of them
(right).
We can apply the same argument to the half of the geodesic closest to z ∈ Ω1 ,
after first substituting ρ = |y − z| − ρ to obtain a similar bound. Combining
these with the original inequality, we have
|f (z) − f (y)|dzdy
Ω2 Ω1
n−1 diam(Ω)
(μ(Ω1 ) + μ(Ω2 )) |∇f (y)|dy.
≤2
n
Ω
To show a local Poincaré inequality on X, we will split the balls, which are
not necessarily convex, up into smaller overlapping convex pieces. We will
do this using the wedges. Because X is admissible, we can use a chaining
argument in order to move through B from one of the Wk to another. We
will say Wk and Wj are adjacent if they share an n − 1-dimensional face, and
we let N (j) be the list of indices of faces adjacent to Wj including j. In order
to create paths which we can integrate over, we need an overlapping region
between the adjacent faces. For k ∈ N (j), let Wk,j = Wj,k be the largest
subset of Wk ∪ Wj which has the property that Wk ∪ Wk,j and Wj ∪ Wk,j
are both convex. Then, for each x in Wk,j there is a way of describing the
rays between x and Wk in a distance preserving manner as one would have
in Rn .
Example 2.4. In Figure 4 we have a complex and ball with two adjacent
wedges. The union of the wedges, W1 and W2 , is not convex, so we form
the region W1,2 . In this example, both W1 ∪ W1,2 and W2 ∪ W1,2 are half
circles.
Theorem 2.5. Let X be a Euclidean polyhedral complex satisfying the
geometric assumptions 1.10. For each x0 ∈ X and 0 < r < R(x0 ) let
B = B(x0 , r), and let its corresponding wedges be labeled Wi,j . The following holds for f ∈ W 1,2 (X) ∩ L1 (B):
n
2 r(μ(Wk ) + μ(Wj,k ))
μ(B)
+2
||∇f ||1,B .
||f − fB ||1,B ≤ 2M max
nμ(Wj,k )
k,j∈N (k) μ(Wk )
Here R(x0 ) = d(x0 , X (D−1) ), where D satisfies x0 ∈ X (D) − X (D−1) . When
x0 ∈ X (0) , R(x0 ) = inf v∈X (0) ,v=x0 d(x0 , v).
474
Melanie Pivarski and Laurent Saloff-Coste
Proof. Note that when x0 is located on a D, but not D−1, dimensional face
the restriction 0 < r < R(x0 ) forces B(x0 , r) to avoid all faces of dimension
D − 1 and lower.
For x in B we have by Jensen:
1
|f (x) − f (y)|dy.
|f (x) − fB | ≤
μ(B) B
We would like to apply Lemma 2.3 to this; however, B is not necessarily
convex. We will construct a path from x to y using a finite number of straight
lines, where each of the line segments is contained in a convex region. For
simplicity, we consider x ∈ Wi and y ∈ Wk . X is locally (n − 1)-chainable,
so there is a chain in B − {x0 } starting at Wi and ending at Wk indexed
by the sequence σ(1) = i, . . . , σ(m) = k, so that for each j, Wσ(j) and
Wσ(j+1) are adjacent, and none of the indices repeat. The path alternates
between wedges and overlapping regions, moving from a point in Wσ(j) into
a connecting point in Wσ(j),σ(j+1) , and then from that connecting point in
Wσ(j),σ(j+1) into a point in Wσ(j+1) . We can take points in these regions:
z1 ∈ Wσ(1) , z2 ∈ Wσ(1),σ(2) , . . . , z2j−1 ∈ Wσ(j) and z2j ∈ Wσ(j),σ(j+1) . Note
that each pair in this sequence is located in a convex region: either
Wσ(j) ∪ Wσ(j),σ(j+1) or Wσ(j+1) ∪ Wσ(j),σ(j+1) .
The line segments between these points define our path γ from x to y.
|f (x) − f (y)| ≤ |f (x) − f (z1 )|
+
l−1
(|f (z2j ) − f (z2j−1)| + |f (z2j ) − f (z2j+1 )|)
j=1
+ |f (z2l ) − f (y)|.
Since this holds for any zi in its corresponding wedge, we can average the
pieces over all of the possible z’s.
|f (x) − f (z1 )|dz1
|f (x) − f (y)| ≤ −
Wi,σ(1)
l−1 +
−
j=1
−
|f (z2j ) − f (z2j−1 )|dz2j dz2j−1
Wσ(j) Wσ(j),σ(j+1)
+−
−
|f (z2j ) − f (z2j+1 )|dz2j dz2j+1
Wσ(j+1) Wσ(j),σ(j+1)
+−
|f (z2l ) − f (y)|dz2l .
Wσ(l),k
We will not keep track of the exact path between every pair of regions,
although in specific examples one may want to do that in order to achieve
475
Small time heat kernel behavior
a tighter bound. Rather, we integrate over all pairs of neighboring wedges.
− |f (x) − f (z)|dz +
− − |f (z) − f (w)|dzdw
... ≤
l∈N (i) Wi,l
j
−
+
l=i,l∈N (j) Wl Wj,l
|f (z) − f (y)|dz.
j∈N (k) Wj,k
This new inequality will hold for x and y in any pair of Wi and Wk with
k = i. If we expand our notation so that Wi,i = Wi , then this will hold
when x and y are in the same set Wk = Wi . To integrate over all y ∈ B, we
can split the integral into two parts; one where x and y are both in Wi and
the second where y is in one of the Wk = Wi . Similarly, we can integrate
over x in Wi and then sum over i.
1
|f (x) − f (y)|dydx
μ(B) B B
⎛
1 ⎝ − |f (x) − f (z)|dzdydx
≤
μ(B)
i,k l∈N (i) Wi Wk Wi,l
− − |f (z) − f (w)|dwdzdydx
+
i,k,j l∈N (j) Wi
+
i,k j∈N (k) Wi
Wk Wj,l Wl
−
⎞
|f (z) − f (y)|dzdydx⎠ .
Wk Wj,k
We first integrate to reduce these to double integrals. We then combine
them into one double sum by setting x = w and y = w as well as reindexing
so that i = j and l = k.
μ(B)
+2
−
|f (z) − f (w)|dzdw.
··· ≤
μ(Wk )
Wk Wj,k
k j∈N (k)
Applying Lemma 2.3 with Ω = Wk ∪ Wj,k , Ω1 = Wj,k , Ω2 = Wk , and
diam(Ω) ≤ 2r to each of the pieces we find:
n
μ(B)
2 r(μ(Wk ) + μ(Wj,k ))
+2
|∇f (y)|dy.
··· ≤
μ(Wk )
nμ(Wj,k )
Wk ∪Wj,k
k j∈N (k)
Note that points in the sets Wk ∪ Wj,k are counted at most 2M times, since
each of the Wk has at most M neighbors. This allows us to combine the
sums to find:
1
|f (x) − f (y)|dydx
μ(B) B B
n
2 r(μ(Wk ) + μ(Wj,k ))
μ(B)
+2
|∇f (y)|dy.
≤ 2M max
nμ(Wj,k )
k,j∈N (k) μ(Wk )
B
476
Melanie Pivarski and Laurent Saloff-Coste
This is the desired result.
We use the uniform bound on the solid angles formed to simplify the
constant in Theorem 2.5.
Corollary 2.6. Let X be a Euclidean polyhedral complex satisfying the geometric assumptions 1.10. For each f ∈ W 1,2 (X) ∩ L1 (B), z ∈ X, and
0 < r < R(z) we have
||f − fB ||1,B ≤ CX r||∇f ||1,B
3n+3
2
where B = B(z, r) and the constant CX = 2 αnM depends only on the
space X. Here R(z) = d(z, X (D−1) ), where D is the dimension such that
z ∈ X (D) − X (D−1) . When z ∈ X (0) , R(z) = inf v∈X (0) ,v=z d(z, v).
μ(Wk )+μ(Wj,k )
μ(B)
+
2
from TheProof. We need to bound maxk,j∈N (k) μ(W
μ(Wj,k )
k)
orem 2.5. Since we want to bound μ(Wj,k ), we need a way to compare it to
μ(Wk ). We can subdivide the space initially by cutting each piece in half
in each of the n dimensions, so that there are at most M = 2n M pieces.
has a volume which
When Wk and Wj are adjacent, this tells us that Wj,k
is larger than min(μ(Wj ), μ(Wk )). Thus
)
μ(Wk )+μ(Wj,k
)
μ(Wj,k
≤ 2.
μ(B)
we will need the solid angle bound. Combining the solid
To bound μ(W
k)
angle bound with the factor of 2−n decrease in wedge size gives us the
modified inequality:
μ(Wk )
.
α
Summing the left-hand side of the inequality over k tells us that
μ(Wk ) ≤ 2−n μ(r n S (n−1) ) ≤
μ(B) ≤ M 2n 2−n μ(r n S (n−1) ).
If we multiply the right-hand side of the inequality by M 2n , we have
M 2n μ(Wk )
.
α
Combining these two inequalities, we find that:
M μ(r n S (n−1) ) ≤
M 2n
μ(B)
≤
.
μ(Wk )
α
We can substitute these into our constant to get:
n
))
2 r(μ(Wk ) + μ(Wj,k
μ(B)
+
2
2M max
)
nμ(Wj,k
k,j∈N (k) μ(Wk )
≤ 2M 2n
n+1
2
r
M 2n
+2
.
α
n
Small time heat kernel behavior
477
This combined with Theorem 2.5 gives us:
||f − fB ||1,B ≤
23n+3 M 2
r||∇f ||1,B .
αn
We would like to extend these theorems so that the radius is not dependent
on the center of the ball. To do so, we show a weak Poincaré inequality for
the Euclidean metric where the bound on the radius is independent of the
ball’s center. We then transfer it to the Riemannian metric and extend it
to a stronger version.
Theorem 2.7. Let X be a Euclidean polyhedral complex satisfying the geometric assumptions 1.10. The following inequality holds for each z ∈ X,
0 < r < R0 , and and f ∈ W 1,2 (X) ∩ L1 (B):
|f (x) − fB(z,r) |dx ≤ rCW
|∇f (x)|dx.
B(z,r)
B(z,κr)
log2 (κ)
Here CW =
23n+3 M 3 κCD
αn
, κ=6
√ 2
2(1−cos(α))
n
+ 1 , CD is the volume
doubling constant for balls of radius less than or equal to , and R0 = κ .
−n
2
−1
√
+1
. If d(z, X (n−1) ) > r,
Proof. Let z ∈ X and r ≤ 6
2(1−cos(α))
then the result follows as a weaker version of Corollary 2.6. Otherwise, we
need to find a point vk which has the property that it is on a k-skeleton, and
there are no other faces in the k-skeleton intersected by B(vk , d(vk , z) + r).
We will do this by descending down the skeleta.
If there is a point within r of z with this property, we will use it.
If not, set r0 = 3r. Then there is a k such that the lowest dimensional
skeleton intersected by B(z, r0 ) is X (k) , and X (k) is intersected by B(z, r0 )
in at least two points, vk and wk , on two different faces. If these faces did
not intersect, then they are at least distance from one another. This would
imply that ≤ 2r0 = 6r, a contradiction. Thus those two faces intersect
in a smaller j-dimensional face. Call vj the point on the j-dimensional face
which minimizes min(d(vj , vk ), d(vj , wk )). These three points form a triangle
with angle vk vj wk ≥ α, where α is the smallest interior angle in X. Note
k)
that this angle is bounded by the assumption α ≤ μ(rμ(W
n S n−1 ) . The triangle
that maximizes min(d(vj , vk ), d(vj , wk )), the minimum distance to this new
point, is an isosceles one with angle vk vj wk = α. The law of cosines tells us
that for the isosceles triangle, d(vk , wk )2 = 2d(vk , vj )2 (1 − cos(α)), and so
.
for a general triangle, d(vk , vj ) ≤ √ d(vk ,wk ) ≤ √ 2r0
2(1−cos(α))
2(1−cos(α))
Otherwise, we have at least two points, vj
If this vj works,
we are done. and wj within √ 2
+ 1 r0 of z. We repeat the process by taking
2(1−cos(α))
478
Melanie Pivarski and Laurent Saloff-Coste
new r’s of the form ri+1 =
√ 2
2(1−cos(α))
+ 1 ri and finding a point on a
lower-dimensional skeleton. The worst case scenario will have us repeat this
n times until we are left with at least one point on the lowest-dimensional
skeleton
that intersectsB(z, R). The largest radius that we could require is
√
n
2
2(1−cos(α))
3r. Using this R, we can show that B(v, R) does
−n
+1
not intersect two such faces. The condition r ≤ 6−1 √ 2
R=
+1
2(1−cos(α))
tells us that R ≤
As two nonintersecting faces cannot be closer than the
closest pair, B(v, R) contains at most one.
This construction
gives us a center, v, on a k-dimensional face, and a
1
2 .
radius, R ≤
√
n
2
2(1−cos(α))
+1
3r so that B(v, R) intersects only the k-
dimensional face that v is on. This allows us first to recenter our ball around
v and then to apply Corollary 2.6 to f on B(v, R). Then, as
n
2
+1 ,
κ=6 2(1 − cos(α))
we find B(v, R) ⊂ B(z, κr).
|f (x)−fB(z,r) |dx
B(z,r)
1
≤
μ(B(z, r))
|f (x) − f (y)|dxdy
B(z,r)
B(z,r)
μ(B(v, R))
−
|f (x) − f (y)|dxdy
≤
μ(B(z, r)) B(v,R) B(v,R)
μ(B(v, R)) 23n+3 M 2
R
|∇f (x)|dx
≤
μ(B(z, r))
αn
B(v,R)
3n+3 M 2
log2 (κ) 2
κr
|∇f (x)|dx.
≤ M CD
αn
B(z,κr)
We can extend the Euclidean case to the Riemannian one via the ellipticity
bounds using metric comparison arguments similar to those in Theorem 5.1
in [13]. Recall that the subscript e refers to Euclidean objects and g refers
to Riemannian ones.
Theorem 2.8. Let X be an admissible n-dimensional Riemannian polyhedral complex with ellipticity bound Λ, Riemannian distance between nonadjacent faces bounded below by Λ, and Euclidean volume bound
μe (Be (z, Λr)) ≤ M Λ2n μe (Be (z, r/Λ))
Small time heat kernel behavior
479
for r < R0 . Suppose f satisfies the following weak Euclidean Poincaré
inequality for any r < R0 :
||f − fBe (z,r) ||1,Be (z,r) ≤ rCW ||∇e f ||1,Be (z,κr).
Then f also satisfies the following weak Riemannian Poincaré inequality for
any r < RΛ0 :
||∇g f ||1,Bg (z,κ r) .
||f − fBg (z,r) ||1,Bg (z,r) ≤ rCW
In the second inequality the norms are with respect to the Riemannian met = Λ6n+2 M C , and κ = Λ2 κ.
ric, CW
W
Proof. We use the comparisons Bg (r) ⊂ Be (Λr) and dμg (x) ≤ Λn dx to
compare the Riemannian lefthand side with a Euclidean version:
||f −fBg (z,r) ||1,Bg (z,r)
1
≤
|f (x) − f (y)|dμg (x)dμg (y)
μg (Bg (z, r)) Bg (z,r) Bg (z,r)
Λ2n
|f (x) − f (y)|dxdy.
≤
μg (Bg (z, r)) Be (z,Λr) Be (z,Λr)
We apply the Euclidean weak Poincaré inequality from Theorem 2.7 to get:
1
|f (x) − f (y)|dxdy
μe (Be (z, Λr)) Be (z,Λr) Be (z,Λr)
|∇e f (x)|dx.
≤ ΛrCW
Be (z,κΛr)
For the right-hand side, we can use the comparisons Be (κΛr) ⊂ Bg (κΛ2 r),
dx ≤ Λn dμg (x), and |∇e f | ≤ Λ|∇g f |:
|∇e f (x)|dx ≤ Λn+1
|∇g f (x)|dμg (x).
Be (z,κΛr)
Bg (z,κΛ2 r)
We can chain these inequalities together to get:
||f − fBg (z,r) ||1,Bg (z,r) ≤
Λ2n μe (Be (z, Λr))
ΛrCW Λn+1 ||∇f ||1,Bg (z,κΛ2 r) .
μg (Bg (z, r))
We can use volume doubling to simplify the constant. We know:
μe (Be (z, Λr)) ≤ M Λ2n μe (Be (z, r/Λ)) ≤ M Λ3n μg (Bg (z, r)).
This gives us:
||f − fBg (z,r) ||1,Bg (z,r) ≤ rM Λ6n+2 CW ||∇fg ||1,Bg (z,κΛ2 r) .
Corollary 2.9. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. Let f ∈ W 1,p (B). For all 0 < r < R0 and
p ∈ [1, ∞), the strong p-Poincaré inequality holds:
pCD ||∇f ||p,Bg (z,r) .
||f − fBg (z,r) ||p,Bg (z,r) ≤ rCW
480
Melanie Pivarski and Laurent Saloff-Coste
Here CD is a constant which depends only on the volume doublingconstant
−n
2
−1
−1
√
+1
, and
for balls of radius less than Λ, R0 = 6 Λ
2(1−cos(α))
CW
is defined as in Theorem 2.8.
Proof. Apply Theorem 2.8 to Theorem 2.7 to get a weak Riemannian 1Poincaré inequality. Then apply Theorem 2.4 from [37] to get a strong
uniform Riemannian 1-Poincaré inequality. Note that the theorem from
[37] involves Poincaré inequalities with 2-norms, changing a weak inequality
where the right-hand side is integrated over a ball of radius 2r into a strong
one; however, the proof generalizes to 1-norms and balls of radius Cr for
some constant C. The 1-Poincaré inequality then becomes a p-Poincaré
inequality by an application of Hölder’s inequality; a calculation gives the
constants. Note that the function space W 1,p (B) is allowed due to the use
of the Whitney cover.
Corollary 2.10. Let X be a Riemannian polyhedral complex satisfying the
pC so
geometric assumptions 1.10. For any R0 there exists a constant CW
D
that for all r < R0 and p ∈ [1, ∞), the strong p-Poincaré inequality holds
for all f ∈ W 1,p (B):
pCD
||∇f ||p,Bg (z,r) .
||f − fBg (z,r) ||p,Bg (z,r) ≤ rCW
is a constant depending only on the volume doubling constant for
Here CD
n
2
2
is defined as
+ 1 , and CW
balls of radius less than 6Λ R0 √
2(1−cos(α))
in Theorem 2.8.
The proof involves successive uses of Whitney covers to expand the radius
for a p = 1 inequality. This gives the dependence on the local volume
doubling constant. This expands to a p-inequality using Hölder. For full
details, see [30].
3. Applications
3.1. Heat kernel bounds. The heat kernel, ht (x, y), is the kernel of the
semigroup e−tΔ as well as the fundamental solution to the heat equation
∂t u = −Δu.
Note that our formulation does not have a factor 12 , which appears in the
probability literature. We include a minus sign as our Laplacian is defined
as a nonnegative operator.
In the present context it is important to state precisely what is meant
by the solution of the Poisson equation Δu = f in an open set Ω or of the
heat equation (∂t + Δ)u = 0 in (a, b) × Ω. Observe indeed that there is no
established notion of a classical solution in this context. The most useful
notion is probably that of a local weak solution. We refer the reader to
Small time heat kernel behavior
481
Sturm [37] which gives a detailed account. In the easier case of the Poisson
equation, a function u is a weak solution of Δu = f in Ω (with f a locally
integrable function, say) if:
1,2
(Ω). • u is Wloc
• Ω ∇u · ∇φdμ = Ω f φ for all φ ∈ W01,2 (Ω).
The definition of weak solution of the heat equation is somewhat more complicated because it involves hypotheses on u and ∂t u. The main hypothesis
on u(t, x) is that it belongs (locally) to the Banach space
L2 ((a, b) → W 1,2 (Ω)).
See Sturm [37]. A simple but crucial fact is that for any f ∈ L2 (X), the function u(t, x) = e−tΔ f (x) is a weak solution of the heat equation on (0, ∞)×X.
When volume doubling and the Poincaré inequality hold true locally, the
function u(t, x) = ht (x, y), y ∈ X fixed, is a weak solution of the heat
equation on (0, ∞) × X, and the time derivatives uk (t, x) = ∂tk ht (x, y) are
also weak solutions of the heat equation on (0, ∞) × X. In this section, we
point out some of the many important consequences of the fact that under the geometric assumptions 1.10, (X, μ, E, Dom(E)) satisfies the uniform
local volume doubling property and the uniform local Poincaré inequality
(Corollary 2.10).
Remark 3.1. If X (k) , the k-skeleton of X, is connected and X satisfies
assumptions 1.10, then X (k) will also satisfy 1.10. In that case, all of the
results stated for X in this section also apply to X (k) (of course, the dimension of X (k) is k and so n should be replaced by k in all of the results). This
occurs, for example, whenever X is comprised solely of polytopes.
Corollary 3.2. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. Then E is conservative. In particular, the
following are true:
• X is stochastically complete: for all x ∈ X and t > 0,
ht (x, y)dμ(y) = 1.
X
• For some α > 0 every nonnegative solution u ∈ L∞ (X, dμ) of
(Δ + α)u = 0
is identically zero.
• For all α > 0 every nonnegative subsolution u ∈ L∞ (X, dμ) of
(Δ + α)u = 0
is identically zero.
Proof. Note that our assumptions on the local geometry of the space give
us uniform local volume doubling bounds. We use these bounds, but not
any Poincaré inequalities, to show the result. The volume doubling bounds
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Melanie Pivarski and Laurent Saloff-Coste
imply via covering arguments (see Lemma 5.2.7 in [31]) that volume grows
at most as an exponential function. That is, there exist constants C0 , C1 so
that for all x ∈ X and all r ≥ 1 the following inequality holds:
μ(B(x, r)) ≤ C0 eC1 (1+r) .
For r ≥ 1, this gives us
r
r
≥
ln(μ(B(x, r)))
ln(C0 ) + C1 (1 + r)
which implies
∞
1
r
dr = ∞.
ln(μ(B(x, r)))
By Theorem 4 in [35] E is conservative.
The heat equation is parabolic; one way of obtaining information about
it is through parabolic Harnack inequalities. Sturm [37] shows that local
volume doubling and Poincaré inequalities on a subset of a complete metric
space imply a local parabolic Harnack inequality on that subset. He then
uses this to find Gaussian estimates on the heat kernel. The equivalence
of the parabolic Harnack inequality with Poincaré and volume doubling
had previously been done in the Riemannian manifold case by Grigor’yan
[17] and Saloff-Coste [32]. We apply Sturm’s estimates to the Riemannian
complex case, first showing the local Harnack inequality and then the heat
kernel bounds.
Corollary 3.3. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. For any R0 > 0 there exists a constant
CH = CH (X, R0 )
so that for any x ∈ X, r ∈ (0, R0 ), and all t > 0 we have:
sup
(s,y)∈Q−
u(s, y) ≤ CH
inf
(s,y)∈Q+
u(s, y)
whenever u is a nonnegative local solution of (Δ + ∂t )u = 0 on the cylinder
Q = (t − 4r 2 , t) × B(x, 2r). Here Q− = (t − 3r 2 , t − 2r 2 ) × B(x, r) and
Q+ = (t − r 2 , t) × B(x, r).
Proof. This follows from Theorem 3.5 in [37]. The constant depends only
on R0 (and not the choice of x, r ∈ (0, R0 )) due to the uniform local volume
doubling and the uniform local Poincaré inequality in Corollary 2.9.
The uniform Harnack inequality gives us uniform Hölder continuity for
local solutions of the heat equation.
Corollary 3.4. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. For R0 > 0 there are constants C = C(X, R0 )
Small time heat kernel behavior
483
and α ∈ (0, 1) so that for any x ∈ X, T ∈ (−∞, ∞) and r < R0 we have:
α
|s − t|1/2 + |y − z|
|u(s, y) − u(t, z)| ≤ C sup |u|
r
Q
whenever u is a local solution of (Δ + ∂t )u = 0 on the cylinder
Q = (t − 4r 2 , t) × B(x, 2r),
s, t ∈ (T − r 2 , T ), and y, z ∈ B(x, r).
√
For any t ∈ (0, ∞), any x, y, z ∈ X with z ∈ B(y, min(1, t)), the heat
kernel and its time derivatives satisfy :
α
d(y, z)
1
j
j
√
h2t (x, y).
∂t ht (x, y) − ∂t ht (x, z) ≤ C
min(1, t)j
min(1, t)
Proof. Apply Proposition 3.1 in [37], noting that we have a uniform constant for the Harnack inequality (Corollary 3.3). To obtain the heat kernel
estimate, use the first statement, analyticity of the heat kernel with respect
to the time variable, and the Harnack inequality for the heat kernel itself. Definition 3.5. A nonnegative solution u is minimal if for any solution v
with the property 0 ≤ v ≤ u, there must exist a constant λ ∈ [0, 1] so that
v = λu.
The Harnack inequality allows us to write global minimal solutions of the
heat equation on (−∞, ∞) × X in terms of minimal solutions to an elliptic
equation.
Corollary 3.6. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. Let u ≥ 0 be a minimal solution to the heat
equation on (−∞, T )×X, where T < ∞. Then there exists a constant C and
a minimal solution f to the equation Δf = Cf so that u(t, x) = eCt f (x).
Proof. Apply the argument of Theorem 2 in Koranyi and Taylor [26] using
the local Harnack inequality from Corollary 3.3.
Theorem 3.7. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. For any R0 > 0 there is a corresponding constant C = C(X, R0 ) so that
C
1
≤ ht (x, x) ≤ n/2 .
n/2
Ct
t
for all x ∈ X and all t such that 0 < t < R02 .
Proof. See, e.g, Theorems 4.1 and 4.3 in [37]. The stated bound easily
follows from the volume estimate c1 r n ≤ μg (B(x, r)) ≤ C1 r n , r ∈ (0, R0 ),
and from the Harnack inequality. See, e.g., [31, Section 5.4.6].
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Melanie Pivarski and Laurent Saloff-Coste
We also have off-diagonal bounds on the heat kernel. To state these in
the most precise form, let us introduce the bottom λ0 of the spectrum of Δ,
that is,
λ0 = inf{E(f, f ) : f ∈ Dom(E), f 2 = 1}.
Corollary 3.8. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. For any R0 there exists C = C(X, R0 ) and
Cj = Cj (X, R0 ) so that for any x, y ∈ X and t > 0 we have the following
bounds:
N/2
2
C
d2 (x, y)
− d (x,y)
−λ0 t
4t
e
1+
ht (x, y) ≤
t
(min(t, R02 ))n/2
2
− Ct2
1
−C d (x,y)
R0
t
e
e
Cμ(B(x, min(t, R02 )))
N/2+j
C (1 + λ t)1+N/2+j d2 (x,y)
d2 (x, y)
j
j
0
− 4t −λ0 t
e
.
1+
∂t ht (x, y) ≤ j
t
t (min(t, R02 ))n/2
Here, N depends only on the volume doubling constant for balls of radius
R0 .
ht (x, y) ≥
Proof. To get the upper bound, for each x, y ∈ X, apply Theorem 4.1 in
[37] to X, taking Y = B(x, R0 )∪B(y, R0 ). Similarly, for each x, y ∈ X apply
Theorem 4.8 from [37] to X with Y = B(x, 2R0 ) to get the lower bound.
To prove the derivative inequality, we show that a local Sobolev inequality
holds. The hypotheses of Theorem 3.7 allow us to use Corollary 2.9 to
find uniform constants for local volume doubling and the local Poincaré
inequality. These give a local Sobolev inequality by Theorem 2.6 in [37]. In
particular, for any R0 > 0 we have both a volume doubling constant and a
Sobolev inequality constant that work for all balls of radius R0 in X. For
the third result, apply Theorem 2.6 from [36] to X.
One of the most interesting applications of the local Harnack inequality
provided by the uniform local volume doubling and local Poincaré inequality is a rather complete description of global positive weak solutions which
includes, in particular, a strong statement concerning uniqueness for the positive Cauchy problem. This application goes back to the work of Aronson.
The generality of the argument is clearly pointed out in [1]. The statement
in the present situation is as follows.
Theorem 3.9. Let X be a Riemannian polyhedral complex satisfying the
geometric assumptions 1.10. There exists a constant C such that any nonnegative weak solution u of the heat equation on (0, T ) × X satisfies
t d(x, y)2
.
∀ x, y ∈ X, 0 < s < t < T, u(s, x) ≤ u(t, y) exp C 1 + +
s
t−s
Moreover, uniqueness of the positive Cauchy problem holds on (0, T ) × X
for the heat equation. More precisely, if u is a nonnegative weak solution
Small time heat kernel behavior
485
of the heat equation on (0, T ) × X, then there exists a unique nonnegative
Borel measure γ on X and a > 0 such that
2
e−ad(x0 ,x) γ(dx) < ∞
X
for some (equivalently, any) x0 ∈ X and
ht (x, y)γ(dy), (t, x) ∈ (0, T ) × X.
u(t, x) =
X
In particular, if u is a nonnegative weak solution of the heat equation on
(0, T ) × X and
∀ f ∈ C0Lip (X), lim
u(t, ·)f dμ =
u0 f dμ
t→0 X
for some u0 ∈
L1
loc (X),
then u(t, x) =
X
X
ht (x, ·)u0 dμ.
An important consequence of Corollary 3.4 and Corollary 3.8 is the following result.
Theorem 3.10. Consider the Laplacian Δ, restricted to the set
D = D0∞ (X) ∩ Dom(Δ)
(see Definition 1.20 and Proposition 1.21) as a densely defined unbounded
operator on the Banach space C0 (X) equipped with the uniform norm · ∞ .
Then (Δ, D) is a closable operator, and its closure is the infinitesimal generator of the strongly continuous semigroup of operators on C0 (X) defined
by
f →
ht (·, y)f (y)dμ(y).
X
Proof. We already noted that D is dense in C0 (X) (see Proposition 1.21).
It is not hard to verify that (Δ, D) satisfies the positive maximal principle.
It follows that, if it is closable, it is the generator of a Feller semigroup
(a strongly continuous, positivity preserving, semigroup of contraction on
C0 (X)). Closability is rather difficult to prove directly, but Corollary 3.4
and Corollary 3.8 show that the heat semigroup, originally defined as an L2
Markov semigroup, actually produces a strongly continuous semigroup on
C0 (X). The infinitesimal generator of this semigroup on C0 (X) is a (closed!)
extension of (Δ, D). This proves that (Δ, D) is closable. The uniqueness
results in [28, 40] show that the semigroup associated with the closure of
(Δ, D) on C0 (X) must coincide with the heat semigroup already constructed.
When the complex X both satisfies a global Poincaré inequality and has a
global volume doubling bound, stronger results follow that do not hold true,
in general, under the weaker hypotheses considered so far. Examples of
complexes satisfying these global assumptions are given in the next section.
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Melanie Pivarski and Laurent Saloff-Coste
Corollary 3.11. Let X be a Riemannian complex which satisfies both a
global 2-Poincaré inequality and volume doubling as well as assumptions 1.10.
There exist N = N (X), C = C(X) and Cj = Cj (X) so that for any x, y ∈ X
and t > 0 we have:
N/2
d2 (x,y)
C
d2 (x, y)
− 4t
√
e
1+
ht (x, y) ≤
t
μ(B(x, t))
d2 (x,y)
1
√ e−C t
ht (x, y) ≥
Cμ(B(x, t))
N/2+j
d2 (x,y)
Cj
d2 (x, y)
j
−j − 4t
√ t e
.
1+
∂t ht (x, y) ≤
t
μ(B(x, t))
Proof. The heat kernel bounds follow directly from Corollaries 4.2 and 4.10
in [37]. The derivative bound follows from Corollary 2.7 in [36].
Corollary 3.12. Let X be a volume doubling Riemannian complex satisfying a global 2-Poincaré inequality as well as assumptions 1.10. When we fix
an arbitrary point x ∈ X we find (E, Dom(E)) is recurrent if and only if
∞
rdr
= ∞.
μ(B(x, r))
1
In the case where (E, Dom(E)) is transient we have the following estimate
for the Green’s function :
∞
∞
∞
rdr
rdr
≤
ht (x, y)dt ≤ C
c
d(x,y) μ(B(x, r))
0
d(x,y) μ(B(x, r))
where c, C are constants depending on X.
Proof. The Poincaré inequality, volume doubling, and the fact that E is
irreducible allow us to apply Corollary 2.9 in [36]. The Poincaré inequality
and volume doubling allow us to apply Corollary 4.11 in [37] to get the
bounds on the Green’s function.
3.2. Groups with polynomial growth. A collection of examples can be
found by considering metric spaces, X, which are acted upon by a finitely
generated group of isometries, G. If X/G = Y and Y can be expressed as a
finite Riemannian complex, then X is a Riemannian complex as well. Note
that the k-skeleton of Y is X (k) /G. We describe the heat kernel behavior
for the complex when the group has polynomial volume growth.
We begin by recalling the relevant definitions for finitely generated groups.
Then we state a proof that any finitely generated volume doubling group
satisfies a Poincaré inequality. We use this to compare functions on the
complex with related functions on the group. This will allow us to give the
heat kernel behavior for the complex.
Definition 3.13. A finite product of elements from a set S is called a word.
If a word is written s1 s2 . . . sk , we say it has length k.
Small time heat kernel behavior
487
Definition 3.14. A finitely generated group is a group with a generating
set, S, where every element in the group can be written as a finite word
using elements of S. Although for a given g ∈ G it can be computationally
difficult to determine which word is the smallest one representing g, such a
word (or words) exists. If this word has length k, then we write |g| = k.
When S = S −1 , we say S is symmetric.
Definition 3.15. We define the volume of a subset of G to be the number
of elements of G contained in that subset. We write |Br | to denote the
volume of a ball of radius r, Br := {g ∈ G : |g| ≤ r}. For groups, volume is
translation invariant, and so we do not lose any generality by calculating the
volume of a ball centered at the identity. A group has polynomial volume
growth when |Br | is bounded above by a polynomial as r tends to infinity.
Definition 3.16. For a function f which maps elements of a group to the
reals, we define the Dirichlet form on 2 (G) to be
E(f, f ) =
1 |f (g) − f (gs)|2 .
|S|
g∈G s∈S
Although we would need to specify directions if we were to define a gradient,
we can define an object which behaves
like the length of the gradient of f
1
on G. We write this as |∇f (x)| = |S| s∈S |f (x) − f (xs)|2 . Notationally,
this means that E(f, f ) = g∈G |∇f (g)|2 .
One can show a Poincaré type inequality on a volume doubling finitely
generated group. The arguments used in this can be found in [9]. We include
a proof for completeness.
Lemma 3.17. Let G be a finitely generated group with symmetric generating
set S. For any f : G → R, the following inequality holds on balls Br :
f − fBr 1,Br ≤
|B2r | 2r |S|∇f 1,B2r .
|Br |
If the group is volume doubling with constant CD , this is a weak Poincaré
inequality on balls for p = 1:
f − fBr 1,Br ≤ 2rCD |S|∇f 1,B2r .
Proof. Let G be a finitely generated group with a symmetric set of generators, S. Let Br be a ball of radius r; for brevity, we will not explicitly write
the center. We can write the norm of f minus its average as follows.
f − fBr 1,Br ≤
1 |f (x) − f (y)|.
|Br |
x∈Br y∈Br
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Melanie Pivarski and Laurent Saloff-Coste
For each y ∈ Br , there exists a g ∈ G with |g| ≤ 2r such that y = xg. We
make this substitution and sum over all g ∈ G with |g| ≤ 2r and xg ∈ Br .
1 1 |f (x) − f (y)| ≤
|f (x) − f (xg)|
|Br |
|Br |
x∈Br y∈Br
x∈Br g:|g|≤2r,xg∈Br
=
1
|Br |
|f (x) − f (xg)|.
g:|g|≤2r x:x,xg∈Br
We will begin with the innermost quantity, and then we simplify the sums.
We can write g = s1 . . . sk as a reduced word with k ≤ 2r. We rewrite the
difference of f at x and xg by splitting the path between them into pieces.
|f (x) − f (xg)| ≤
|g|
|f (xs1 . . . si−1 ) − f (xs1 . . . si )|.
i=1
We fix g and sum over all x such that x, xg ∈ Br .
|g|
|f (x) − f (xg)| ≤
|f (xs1 . . . si−1 ) − f (xs1 . . . si )|
x:x,xg∈Br i=1
x:x,xg∈Br
=
|g|
|f (xs1 . . . si−1 ) − f (xs1 . . . si )|.
i=1 x:x,xg∈Br
We can change variables by letting z = xs1 . . . si−1 . If i − 1 ≤ r, then
|xs1 . . . si−1 | ≤ 2r; otherwise, k + 1 − i ≤ r and so
−1
−1
−1
|xgs−1
k . . . si | = |xs1 . . . sk sk . . . si | = |xs1 . . . si−1 | ≤ 2r.
Thus z ∈ B2r .
··· ≤
|g|
|f (z) − f (zsi )|.
i=1 z∈B2r
Since si ∈ S, we can expand the sum to all s ∈ S. To do this, we must
account for the multiplicity of the si . We could have at most |g| copies of
any generator; |g| ≤ 2r, and so we gain a factor of 2r.
|f (z) − f (zs)|.
· · · ≤ 2r
s∈S z∈B2r
Jensen’s inequality allows us to rewrite this in terms of the gradient.
1 |S|
|f (z) − f (zs)|2
. . . ≤ 2r
|S|
z∈B2r
s∈S
= 2r
|S||∇f (z)|.
z∈B2r
Small time heat kernel behavior
489
We use this calculation to get the desired inequality. We take this inequality,
divide by |Br |, and sum over the g to get
1
1
|f (x) − f (xg)| ≤
2r |S||∇f (z)|.
|Br |
|Br |
g:|g|≤2r x∈Br
This reduces to
f − fBr 1,Br ≤
g:|g|≤2r z∈B2r
|B2r | 2r |S|∇f 1,B2r .
|Br |
2r |
Note that in general, |B
|Br | will depend on the radius, r. If the group is
volume doubling, this gives us a weak Poincaré inequality.
f − fBr 1,Br ≤ CD 2r |S|∇f 1,B2r .
Consider a complex, X, and a finitely generated group of isomorphisms,
G, on the complex such that X/G = Y is an admissible complex consisting
of a finite number of polyhedra. One example of this type of complex is
the metric version of a Cayley graph; this is the graph where each vertex
corresponds to a group element, and two vertices are connected by an edge
if they differ by an element of the generating set. In this case, Y is the unit
interval.
The local volume doubling and local uniform Poincaré inequality on X
lead to the easy transfer from G to X of the global volume doubling and
global Poincaré inequality using ideas that go back to Kanai’s work [24]. We
compare functions defined on the complex, X, with functions defined on the
group, G, by transferring a function defined on X to a function defined on
G that roughly preserves the norm of both the function and its energy form.
As we are frequently switching between X and G, we will use BX for balls
in X and BG for balls in G.
Definition 3.18. For functions f : X → R we define f : G → R by
1
f (x)dx = −
f (x)dx.
f (g) =
μ (BX (g, δ)) BX (g,δ)
BX (g,δ)
Here, all integration is with respect to the Euclidean norm and δ := diam(Y ).
It is important to note that the sets {g : g ∈ BX (r) ∩ G} and BG (r) are
potentially different; however, there exist constants such that
r ⊂ BG (r) ⊂ G ∩ BX (C0 r).
G ∩ BX
C
The proof of this can be found in [30].
We can compare the norm of f with the norm of f, as well as the norm of
∇f with that of its analogue. Note that given a radius, R0 , Corollary 2.10
tells us that we have a uniform Poincaré inequality for f on balls of radius
at most R0 . We use this with R0 = 3 diam(Y ) where CP is the constant
490
Melanie Pivarski and Laurent Saloff-Coste
associated to the 1-Poincaré inequality on X. The proofs from [10] for these
inequalities carry over to this case; careful calculations yield the constants.
This type of argument can also be found in Barlow, Bass, and Kumagai [3].
Lemma 3.19. Let BX (r) := BX (g , r) be a ball in X centered at g ∈ G.
For any c ∈ R, r ≥ δ = diam(Y ), f ∈ W 1,1 (X) and corresponding f, we
have the following comparison:
||f − c||1,BX (r) ≤ C ||∇f ||1,BX (3r) + ||f − c||1,BG (2C r) .
||∇f||1,BG (r) ≤ C ||∇f ||1,BX (3C0 r) .
Here, C = (2CP maxx∈X #{g ∈ G∩X|x ∈ BX (g, 2δ)}+2μ(BX (e, δ)))(δ+1),
X (e,2δ))
and C = μ(B
μ(BX (e,δ))2 maxx∈X #{g ∈ G ∩ X|x ∈ BX (g, 2δ)}CP 2δ. Note that
the constants C and C depend on X and Y , but not on r or g .
These bounds can be used to transfer inequalities between X and G.
We can combine them with the weak Poincaré inequality on G to get an
inequality on X. Whenever G has polynomial volume growth, X admits a
Poincaré inequality with uniform constant at all scales.
Theorem 3.20. Let X be an admissible Riemannian complex and G be
a finitely generated group with with polynomial volume growth such that
X/G = Y is a finite polytopal complex satisfying assumptions 1.10. For
1 ≤ p < ∞ there exists a constant C = C(X) so that for all f ∈ W 1,p (X),
all x ∈ X, and all r > 0 we have:
inf ||f − c||p,BX (x,r) ≤ Cr||∇f ||p,BX (x,r) .
c
Note that this implies:
||f − fBX (r) ||p,BX (x,r) ≤ 2Cr||∇f ||p,BX (x,r) .
The constant C does not depend on the center or on the radius of the ball.
Proof. Note that we chose R0 so that the Poincaré inequality holds for balls
of radius up to 3δ := 3 diam(Y ). We need to show that it also holds for balls
of radius greater than 3δ. Let r ≥ 3δ be given. To start, we assume that the
center of BX (x, r) is in G. By choosing a value of c, we obtain something
at least as large as the infimum:
inf ||f − c||1,B (x,r) ≤ ||f − fB (x,2C r) ||1,B (x,r) .
c
X
G
X
We can combine Lemma 3.19 with the weak Poincaré inequality on groups
(Lemma 3.17) to get:
inf ||f − c||1,BX (x,r) ≤ C ||∇f ||1,BX (x,3r) + ||f − fBG (x,2C r) ||1,BG (x,2C r)
c
≤ C ||∇f ||1,BX (x,3r) + 2C rCD |S|||∇f||1,BG (x,4C r)
≤ C(1 + 2C CD |S|C )r||∇f ||1,BX (x,12C0 C r) .
Small time heat kernel behavior
491
By using the triangle inequality and Jensen’s inequality, it can be shown
that
||f − fBX (x,r) ||1,BX (x,r) ≤ 2 inf ||f − c||1,BX (x,r) .
c
If the center, x, were not in G, there is some g ∈ G such that the center is
within δ of g . That is, dX (x, g ) ≤ δ. By inclusions of balls, we know that:
inf ||f − c||1,BX (x,r) ≤ inf ||f − c||1,BX (g ,2r)
c
c
||∇f ||1,BX (g ,24C0 C r) ≤ ||∇f ||1,BX (x,25C0 C r) .
This tells us that X admits a weak 1-Poincaré inequality for the Euclidean
structure. Note that the fact that X/G is the finite polytopal complex Y
forces X to have the same bounds as Y for the edge lengths, angles, and
ellipticity constant. Similarly, we have a bound on the number of edges that
share a vertex because Y is finite and the group G is finitely generated. Thus
X satisfies the hypotheses of Theorem 2.8. Apply Theorem 2.8 to transfer
the weak Euclidean 1-Poincaré inequality to the Riemannian structure; this
yields a weak uniform Riemannian 1-Poincaré inequality. It is extended to
a strong uniform p-Poincaré inequality via Theorem 2.4 in [37] and Hölder’s
inequality.
In [41] Varopolous showed that groups with polynomial growth of degree
d have on diagonal behavior h2n (e, e) ≈ n−d/2 . We show that a similar result
holds for volume doubling complexes with underlying group structure.
Theorem 3.21. Let X be a volume doubling Riemannian complex and G
be a finitely generated group with with polynomial volume growth such that
X/G = Y is a finite polytopal complex satisfying assumptions 1.10. There
exists a constant C = C(X) so that for all x, y ∈ X and t > 0, X satisfies
the on-diagonal heat kernel estimates:
C
1
√
√ .
≤ ht (x, x) ≤
Cμ(B(x, t))
μ(B(x, t))
X additionally satisfies the following off-diagonal estimates where N is the
volume doubling constant. There exist C = C(X) and Cj = Cj (X) so that
for any x, y ∈ X and t > 0 we have:
N/2
d2 (x,y)
C
d2 (x, y)
− 4t
√
e
1+
ht (x, y) ≤
t
μ(B(x, t))
d2 (x,y)
1
√ e−C t
ht (x, y) ≥
Cμ(B(x, t))
N/2+j
d2 (x,y)
Cj
d2 (x, y)
j
−j − 4t
√ t e
.
1+
∂t ht (x, y) ≤
t
μ(B(x, t))
Proof. Apply Theorem 3.7 and Corollary 3.11, noting that by Theorem 3.20
X satisfies both volume doubling and a Poincaré inequality at all scales
uniformly.
492
Melanie Pivarski and Laurent Saloff-Coste
3.3. Further remarks. One related question is how do ∇ and Δ−1/2 compare in Lp ? When p = 2, equality follows from integration by parts. In
the case where X is a manifold and G is a group with polynomial volume
growth, Dungey [12] showed that the Riesz transform, ∇Δ−1/2 is bounded
in Lp for all 1 < p < ∞. Ishiwata [23] expanded this to the discrete case
for nonbipartite covering graphs whose covering transformation group has
polynomial volume growth. Recently Auscher and Coulhon [2] have shown
connections between Riesz transforms and Poincaré inequalities on manifolds. It would be interesting to see whether these ideas transfer over to the
complexes considered in this section.
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Texas A&M University, College Station, TX 77843 USA
pivarski@math.tamu.edu
Cornell University, Ithaca, NY 15843 USA
lsc@math.cornell.edu
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